In this article we use weighted Sobolev spaces for the unknown function \(u=(u^1,\cdots ,u^{d_1})\)  and the inputs \(f^k\) .
The need to introduce weights comes from, for instance, the theory of stochastic partial
differential equations (SPDEs) or stochastic partial differential systems (SPDSs), where a Hölder space approach does not allow us to obtain results of reasonable generality and Sobolev spaces without weights are trivially inappropriate (see [1]} for details). To study such stochastic systems one has to develop a nice regularity theory for the corresponding deterministic systems in advance.
Also Sobolev spaces with weights are very useful in treating degenerate elliptic and parabolic equations (see, for instance, [2]}) and
in studying equations defined on non-smooth domains such as domains with wedges (see, for instance, [3]}, [2]}, [5]}).
